Let \(r_1,r_2,r_3,\ldots\) be a sequence of all rational numbers in \( (0,1) \) except finitely many numbers. Let \(r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots\) be a decimal representation of \(r_j\). (For instance, if \(r_1=\frac{1}{3}=0.333333\cdots\), then \(a_{1,k}=3\) for any \(k\).)
Prove that the number \(0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots\) given by the main diagonal cannot be a rational number.
Let \(m\) and \(n\) be odd integers. Determine \[ \sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.\]
Prove that if a finite ring has two elements \(x\) and \(y\) such that \(xy^2=y\), then \( yxy=y\).
Let \(n\) be a fixed positive integer. Find all functions \( f:\mathbb{R}\to\mathbb{R}\) satisfying \[ f(x^{n+1}-y^{n+1})=(x-y)[f(x)^n+f(x)^{n-1}f(y)+\cdots+f(x)f(y)^{n-1}+f(y)^n].\]
Determine all continuous functions \(f:(0,\infty)\to(0,\infty)\) such that \[ \int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx\] for all \(t>0\).
Determine all nonnegative functions f(x,y) and g(x,y) such that \[ \left(\sum_{i=1}^n a_i b_i \right)^2 \le \left( \sum_{i=1}^n f(a_i,b_i)\right) \left(\sum_{i=1}^n g(a_i,b_i)\right) \le \left(\sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n b_i^2\right)\] for all reals \(a_i\), \(b_i\) and all positive integers n.
Let A be a finite set of complex numbers. Prove that there exists a subset B of A such that \[ \bigl\lvert\sum_{z\in B} z\bigr\lvert \ge \frac{ 1}{\pi}\sum_{z\in A} \lvert z\rvert.\]
Let f be a continuous function from [0,1] such that f([0,1]) is a circle. Prove that there exists two closed intervals \(I_1, I_2 \subseteq [0,1]\) such that \(I_1\cap I_2\) has at most one point, \(f(I_1)\) and \(f(I_2)\) are semicircles, and \(f(I_1)\cup f(I_2)\) is a circle.
Determine all Platonic solids that can be drawn with the property that all of its vertices are rational points.
Let M be an n⨉n matrix over the reals. Prove that \(\operatorname{rank} M=\operatorname{rank} M^2\) if and only if \(\lim_{\lambda\to 0} (M+\lambda I)^{-1}M\) exists.